The anharmonic Larkin Model

PURRELLO, VÍCTOR H.; IGUAIN, JOSÉ L; KOLTON, ALEJANDRO B.

Buenos Aires

Conferencia; Statphys27; 2019

The study of elastic interfaces in random media is relevant for understanding generic properties displayed by a variety of experimental systems, and to successfully classify them into universality classes. A simple approach to capture the disorder-elasticity interplay is the well known Larkin model, in which the elastic part of the energy is harmonic, proportional to the squared gradient of the interface position, (∇u)2, and the disordered medium consists of a random force, uncorrelated for each element of the interface, but constant in the displacement direction. Despite its simplicity, the Larkin model is fundamental to understand the emergence of self-similar geometries, pinning, and to set the natural physical units for the full model. However, in the one-dimensional case, this model becomes super-rough: the interface is self-affine with an exponent ζ > 1, which implies that the harmonic elastic approximation must break down in the thermodynamic limit. In this work, we extend the Larkin model by considering a higher-order elasticity as anharmonic contributions proportional to (∇u)2n , with n > 1 an integer. By heuristic scaling arguments, we obtain the global roughness exponent ζ, the dynamic exponent z, and the harmonic to anharmonic crossover length scale, for arbitrary dimension d and n, yielding an upper critical dimension dc(n) = 4n. We find a precise agreement with numerical calculations in d = 1, where we observe, however, an anomalous ?faceted? scaling. The spectral roughness exponent ζs satisfies ζs > ζ > 1 for any finite n > 1, hence invalidating the usual single-exponent scaling for two-point correlation functions. We show that such d = 1 case is directly related to a family of Brownian functionals parameterized by n, ranging from the random-acceleration model for n = 1, to the Lévy arcsine-law problem for n = ∞. Our results may be experimentally relevant for describing the roughening of nonlinear elastic interfaces in a Matheron-de Marsilly type of random flow.